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https://mathoverflow.net/questions/375179 | 1 | Let $s\in (0,1)$, how i can solve the equation:
$$ (-\Delta)^su=1,\quad\text{in}\quad(-1,1)?$$
I have no idea, any help would be appreciated.
| https://mathoverflow.net/users/167027 | Solving an equation with fractional laplacian | The solution to
$$(-\Delta)^s u(x)=1,\;\;-1<x<1,
$$
with $u(x)=0$ for $|x|\geq 1$ is
$$
u(x)=\frac{\sqrt{\pi}(1-x^2)^s }{4^{s}\Gamma(\tfrac{1}{2}+s)\Gamma(1+s)},$$
as follows from the [integral definition](https://en.wikipedia.org/wiki/Fractional_Laplacian) of the fractional Laplacian.
$$\text{The $n$-dimensional ge... | 2 | https://mathoverflow.net/users/11260 | 375208 | 156,569 |
https://mathoverflow.net/questions/375200 | 0 | **Please note:** *This question has been edited after it became clear from Christian Remling's answer that the original formulation was far from what I really meant to ask.*
Let $T\ne 0$ be a self-adjoint operator on a Hilbert space $H$, with spectrum $\sigma(T)$. For any $x∈H$, denote by $μ\_x$ the spectral measure ... | https://mathoverflow.net/users/127070 | Detecting isolated eigenvalues from local spectral measures | **Updated answer:**
For $k=2$ the conditions are contradictory.
We have the decomposition $H=H\_0\oplus H\_1$. You now impose the following conditions: (1) $TH\_0=H\_1$; (2) $T$ injective on $H\_0$ and $H\_1$; (3) $T^2 H\_0=H\_0$; (4) $Tu=0$ for some $u\notin H\_1$.
Since $N(T^2)=N(T)$ for the self-adjoint operator... | 0 | https://mathoverflow.net/users/48839 | 375214 | 156,570 |
https://mathoverflow.net/questions/375212 | 4 | Let $w=x\_0 x\_1 x\_2 \ldots$ be an infinite word, where each $x\_i\in \{0,1\}$. For each positive integer $k$ (thought of as the jump size of an arithmetic progression) and each residue $0\leq a \leq k-1$ we can form the new "arithmetic progression" word $w\_{a\ {\rm mod}\ k}=x\_a x\_{a+k} x\_{a+2k}\ldots$.
**Questi... | https://mathoverflow.net/users/3199 | Binary words that are nonconstant on long arithmetic progressions | Any Sturmian word will work for Question 1. Before I prove this, I can't resist giving the standard example: the Fibonacci word. The Fibonacci word is defined as the fixed point of the iterative procedure which replaces every $0$ with the string $01$, and replaces every $1$ with the string $0$:
$010010100100101001010... | 8 | https://mathoverflow.net/users/2363 | 375217 | 156,571 |
https://mathoverflow.net/questions/360834 | 32 | The [LMFDB](https://www.lmfdb.org/) describes the elliptic curve [11a3](https://www.lmfdb.org/EllipticCurve/Q/11/a/3) (or 11.a3) as "The first elliptic curve in nature". It has minimal Weierstraß equation
$$
y^2 + y = x^3 - x^2.
$$
My guess is that there is some problem in Diophantus' *Arithmetica*, or perhaps some oth... | https://mathoverflow.net/users/4177 | Why is this "the first elliptic curve in nature"? | I asked Kevin Buzzard to ask John Coates directly, and it's basically as people have surmised: the moniker is due to the fact the curve appears first in Cremona's book as it has the smallest possible conductor, and it has the smallest coefficients. It is *not* due to historical priority, as Coates knows of 8th/9th cent... | 6 | https://mathoverflow.net/users/4177 | 375218 | 156,572 |
https://mathoverflow.net/questions/375226 | 6 | This might be a weird/stupid question, but it came to me a couple of times, and I would like to get an answer for that.
In some papers I read, constantly the authors define some analytic subspaces, say $X$ and $Y$, and then the authors take the intersection product of their cycles $[X]\cdot [Y]$ in the homology group... | https://mathoverflow.net/users/98788 | Intersection theory in analytic geometry | You don't say what kind of space $X$ and $Y$ are subspaces of. But if they sit in an oriented manifold there's an easy way to define an intersection product in homology. Namely if $M$ is an oriented $d$-manifold then there is a Poincaré duality isomorphism
$$ H\_i(M,\mathbf Z) \cong H^{d-i}\_c(M,\mathbf Z)$$
between ho... | 10 | https://mathoverflow.net/users/1310 | 375228 | 156,573 |
https://mathoverflow.net/questions/375177 | 18 | Waldhausen's A-theory is a version of algebraic K-theory of spaces. Concretely, for a (pointed) space $X$, he considers the 'Waldhausen category' $\mathcal R\_f(X)$ of finite retractive CW-complexes over $X$, applies his $S\_{\bullet}$ construction to it, and obtains an infinite loop space, $A(X)$. The functor $A$ is e... | https://mathoverflow.net/users/14233 | On the definition of A-theory | Since the question remains unanswered, let me copy Tom Goodwillie's comment:
>
> If you allow finitely dominated instead of finite, it changes only π0. Analogously, in defining K(R) if you use finitely generated projective modules instead of free, it changes only π0. I believe that this is discussed somewhere in Wa... | 7 | https://mathoverflow.net/users/43054 | 375235 | 156,575 |
https://mathoverflow.net/questions/374382 | 6 | For the ***complete extraction of the factor*** $p$ and its powers from a natural number $n$
let's ***define*** the notation $$ \{n\}\_p := { n \over p^{\nu\_p(n)}} \tag 1$$
$ \qquad \qquad $ *Here $\nu\_p(n)$ means the p-adic valuation of $n$ with respect to prime $p$.*
While looking at the question of existences... | https://mathoverflow.net/users/7710 | When is $\{b^2 - \{b-1\}_2\}_2=1$ with odd $b$? (The bracket-notation explained below) | The question is equivalent to finding all squares which are of the form $2^a - 2^b + 1$. [This MO answer](https://mathoverflow.net/questions/126284/binary-expansion-of-squares) discusses the related question which squares are of the form $2^a + 2^b + 1$, and presumably the techniques in the paper (which admittedly I ha... | 3 | https://mathoverflow.net/users/88679 | 375245 | 156,579 |
https://mathoverflow.net/questions/375135 | 3 | If you consider hyperbolic $n$-space $H^n$, modeled by the open unit ball $B^n \subset \mathbb{R}^n$, then given any two distinct points $x\_1$, $x\_2$ in $H^n$, there is a natural way of identifying the unit tangent spheres $S\_{x\_1}$ and $S\_{x\_2}$ at $x\_1$ and $x\_2$ respectively. Start at $x\_1$. Given a unit ta... | https://mathoverflow.net/users/81645 | What is the name of this geometric structure, where we identify each sphere of vision with the sphere at infinity? | This notion of 'the sphere at infinity' is commonly encountered in hyperbolic geometries. Gromov, in particular, has used it in studying the behavior of discrete transformation groups on hyperbolic manifolds and you might also look at the works of Biquard on prescribing the geometry of the boundary at infinity of an Ei... | 1 | https://mathoverflow.net/users/13972 | 375248 | 156,580 |
https://mathoverflow.net/questions/332958 | 17 | The *height* $ht(\alpha)$ of a positive root $\alpha$ in a (finite, crystallographic) root system $\Phi$ is $\sum\_{i=1}^n c\_i$ where $\alpha = \sum\_{i=1}^n c\_i \alpha\_i$ is its decomposition as a sum of simple roots. I am interested in the polynomial
$$R\_{\Phi}(x):=\sum\_{\alpha \in \Phi^+} x^{ht(\alpha)-1}.$$
... | https://mathoverflow.net/users/33089 | Irreducibility of root-height generating polynomial | (Turning Gjergji Zaimi's comment into a community wiki answer.)
A problem equivalent to the case of $R\_{A\_n}$ is discussed in [Classes of polynomials having only one non-cyclotomic irreducible factor](http://doi.org/10.4064/aa-90-2-121-153), by Borisov, Filaseta, Lam, and Trifonov (*Acta Arithmetica* **90** (1999),... | 8 | https://mathoverflow.net/users/3106 | 375253 | 156,584 |
https://mathoverflow.net/questions/375111 | 3 | Is there a Riemannian metric $\tilde g$ on $\mathbb R^d$ such that
$$\tag{1}
\Delta\_{\tilde g}=e^f(\Delta +1),$$
for some $f\in C^\infty(\mathbb R^d)$? Here $\Delta=\partial\_{x\_1}^2+\ldots+\partial\_{x\_d}^2.$ (Answer: no, as (1) fails on constants. See Edit below).
If there is such a $\tilde g$, it cannot be conf... | https://mathoverflow.net/users/13042 | Is there a metric on Euclidean space that turns the Helmholtz equation into the Laplace equation? | After Terry Tao's comments, I came to the conclusion that the only possible choice of a metric $\tilde g$ and of an operator
$$T\_{\tilde g}= \Delta\_{\tilde g} + \text{lower order terms} $$
that will give
$$\tag{1} T\_{\tilde g}=e^f(\Delta +1)$$
is the following, in Cartesian coordinates:
$$\tag{2}
\tilde g\_{ij}= e^{... | 0 | https://mathoverflow.net/users/13042 | 375254 | 156,585 |
https://mathoverflow.net/questions/374910 | 11 | So I'm doing research in control theory and I have been stuck with this problem for a while. Let me explain my issue, then my proposal, and finally my concrete question.
**Problem:** I have $m<n$ real $n\times n$ positive definite matrices $P\_1,\dotsc,P\_m$. These define ellipsoids $E\_i=\{x\in\mathbb{R}^n\mathrel:x... | https://mathoverflow.net/users/166253 | Show that these vectors are linearly independent almost surely | We denote $L\_{\epsilon}(x):=\{(P\_{1}+\epsilon\_{1})x,\cdots,(P\_{m}+\epsilon\_{m})x\text{ linearly independant}\}$
First we have that for any fixed $x\in\mathbb{R}^{n}$, $$\mathbb{P}(L\_{\epsilon}(x))=0.$$
Indeed if you only consider the $m$ first entries of these $m$ vectors you have get an $m\times m$ random matr... | 1 | https://mathoverflow.net/users/99045 | 375268 | 156,589 |
https://mathoverflow.net/questions/375242 | 6 | Consider the Cuntz algebra $\mathcal{O}\_n$ with $n \geq 2$ and let $\text{End}(\mathcal{O}\_n)$ be the set of all (unital) $\ast$-endomorphisms of $\mathcal{O}\_n$. I was wondering if there exists an element $x \in \mathcal{O}\_n$ such that the evaluation map $\text{End}(\mathcal{O}\_n) \rightarrow \mathcal{O\_n},$ $\... | https://mathoverflow.net/users/64444 | Endomorphisms of the Cuntz algebra | This is true: $\mathcal O\_n$ is singly generated, i.e. there exists $x\in \mathcal O\_n$ such that $C^\ast(x) = \mathcal O\_n$. In particular, if $\phi, \psi \colon \mathcal O\_n \to B$ are $\ast$-homomorphisms such that $\phi(x) = \psi(x)$, then $\phi = \psi$.
There might be a very direct way of showing this, but h... | 10 | https://mathoverflow.net/users/126109 | 375270 | 156,590 |
https://mathoverflow.net/questions/374767 | 5 | By the work of Hill-Yarnall, for the group $G=C\_p,$ all the slices for any spectrum, in particular, for $S^V \wedge H\underline{\mathbb{Z}}$, are classified. Here $V$ is a representation of $C\_p.$
Again, following Yarnall's work, we know the spectrum $S^n \wedge H\underline{\mathbb{Z}}$ has the $n$-slice of the for... | https://mathoverflow.net/users/45223 | Slices for certain $C_p$-spectrum | This follows from the Hill-Yarnall formula for slices (I guess they do *regular* slices, so you have to deal with a shift if you want the classical ones). The reason is as follows: the slice of $X$ in dimension $n$ is given by first applying some algebraic procedure to $\pi\_WX$ where $W$ is a certain representation of... | 3 | https://mathoverflow.net/users/6936 | 375277 | 156,593 |
https://mathoverflow.net/questions/375275 | 14 | Suppose $ZFC$ is consistent. Is there a model $\mathcal{M}$ of $ZFC$ and formulae $\varphi\_D(x)$ and $\varphi\_\in(x,y)$ that define (in $\mathcal{M}$) the domain and membership relation of a model $\mathcal{N}$ of $ZFC$ such that we have $\mathcal{N}\cong\mathcal{M}$ externally (i.e., in $V$) and yet $\mathcal{M}$ th... | https://mathoverflow.net/users/9324 | Is there a model of ZFC that can define a "longer" model of ZFC to which it is isomorphic? | The answer is yes.
Let $M$ be a countable computably saturated model of ZFC with a measurable cardinal $\kappa$. Let $N$ be the Ord-length iterated ultrapower of a measure on $\kappa$. The model $M$ thinks $N$ is a definable well-founded class structure, which is strictly taller than $M$. But $M$ and $N$ are two coun... | 13 | https://mathoverflow.net/users/1946 | 375279 | 156,594 |
https://mathoverflow.net/questions/375278 | 2 | I am looking for a seventh degree polynomial with integer coefficients, which has the following roots.
$$x\_1=2\left(\cos\frac{2\pi}{43}+\cos\frac{12\pi}{43}+\cos\frac{14\pi}{43}\right),$$
$$x\_2=2\left(\cos\frac{6\pi}{43}+\cos\frac{36\pi}{43}+\cos\frac{42\pi}{43}\right),$$
$$x\_3=2\left(\cos\frac{18\pi}{43}+\cos\frac{... | https://mathoverflow.net/users/135040 | Minimal polynomial in $\mathbb Z[x]$ of seventh degree with given roots | In [SageMath](https://sagecell.sagemath.org/), you can enter the following:
```
U.<zeta> = CyclotomicField(43)
P.<x> = PolynomialRing(U)
def c(j): # cos(j * pi / 43)
return (zeta ** j + zeta ** (-j))/2
x1 = 2*(c(2) + c(12) + c(14))
x2 = 2*(c(6) + c(36) + c(42))
x3 = 2*(c(18) + c(22) + c(40))
x4 = 2*(c(20) + c... | 6 | https://mathoverflow.net/users/2530 | 375282 | 156,596 |
https://mathoverflow.net/questions/375284 | 3 | Suppose a function $G:\mathbb{R}^d\rightarrow\mathbb{R}$ is given.
What are some necessary or sufficient conditions on $G$ for there to exist a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and a jointly-measurable function $V:\Omega\times\mathbb{R}^d\rightarrow \mathbb{R}$ that realizes a spatially homogenous ... | https://mathoverflow.net/users/137798 | Existence of Gaussian random field with prescribed covariance | The necessary and sufficient condition is for the function $G$ to be positive definite, that is, for the matrix $(G(x\_j-x\_k))\_{j,k\in[n]}$ to be positive definite for any natural $n$ and any distinct $x\_1,\dots,x\_n$ in $\mathbb R^d$, where $[n]:=\{1,\dots,n\}$.
| 5 | https://mathoverflow.net/users/36721 | 375285 | 156,598 |
https://mathoverflow.net/questions/375290 | 3 | Let $\Sigma$ be the class of univalent (injective) holomorphic functions on $\mathbb{C}\backslash \mathbb{D}$ where $\mathbb{D}$ is the closed unit disk. Analogous to the famous Bieberbach conjecture is the problem of finding sharp bounds for the coefficient $b\_n$ of the functions $$g(z) = z + b\_0 + b\_1 z^{-1} + b\_... | https://mathoverflow.net/users/167930 | Coefficient problem in the class $\Sigma$ | The latest results are not very recent. Most of them are mentioned in the paper
>
> MR1162188 Carleson, Lennart; Jones, Peter W. On coefficient problems for univalent functions and conformal dimension. Duke Math. J. 66 (1992), no. 2, 169–206.
>
>
>
Let $B\_n=\sup\_\Sigma |b\_n|$. The question is about the orde... | 3 | https://mathoverflow.net/users/25510 | 375299 | 156,599 |
https://mathoverflow.net/questions/375292 | 8 | [Reed's conjecture](https://www.openproblemgarden.org/op/reeds_omega_delta_and_chi_conjecture) says that $\chi(G)\leq \lceil\frac{\omega(G)+\Delta(G)+1}{2}\rceil$. One can think of $\lceil\frac{\omega(G)+\Delta(G)+1}{2}\rceil$ as the (rounded-up) average of the trivial lower bound and the trivial upper-bound on $\chi(G... | https://mathoverflow.net/users/17798 | Replacing maximum degree with degeneracy in Reed's conjecture | Here's a reasonably simple counterexample.
Take $C\_9$, and label its vertices $v\_0, \ldots, v\_8$ along the cycle. Let $\mathcal{I}$ be the family of all independent sets of $C\_9$ of size $3$. $\chi(C\_9) = 3$, further:
**Lemma.** For any 3-coloring of $C\_9$ there exists $I \in \mathcal{I}$ with vertices of all... | 7 | https://mathoverflow.net/users/106512 | 375307 | 156,601 |
https://mathoverflow.net/questions/375303 | 14 | (1) Are there formulae $\varphi\_D(x)$ and $\varphi\_\in(x,y)$ defining an internal model $\mathcal{N}$ of $ZFC$ where $\mathcal{N}$ is not set-like and no definable, set-like, internal model $\mathcal{M}$ is elementary equivalent to $\mathcal{N}$?
(2) Are there formulae $\varphi\_D(x)$ and $\varphi\_\in(x,y)$ defini... | https://mathoverflow.net/users/9324 | Is it possible to define an internal model of ZFC which is not set-like and which is not elementary equivalent to any definable set-like model? | Let me describe another method for making long models definable.
First consider the case that $\kappa$ is inaccessible in $L$ and $\lambda$ is a worldly cardinal in $L$ above $\kappa$. Let $G$ be $L$-generic for the forcing to collapse $\lambda$ to $\kappa$. In the forcing extension, there is a set $E\subset\kappa$ t... | 12 | https://mathoverflow.net/users/1946 | 375312 | 156,603 |
https://mathoverflow.net/questions/375267 | 3 | Let $K$ be a non-archimedean field of char 0 and a morphism $f:V \rightarrow W$ of normed $K$-vector spaces given. The map $f$ is said to be strict if $V/\ker(f)$ with the quotient topology is homeomorph to $\mathrm{im}(f)$ with subspace topology.
I read without a proof that if $\mathrm{im}(f)$ is closed then $f$ is ... | https://mathoverflow.net/users/135674 | Sufficent condition for strict morphism of normed vector spaces | Counterexamples are trivially obtained by taking a stricly coarser norm on a Banach space, e.g., the $\ell\_\infty$-norm on $\ell\_2$. For $V=(\ell\_2,\|\cdot\|\_2)$, $W=(\ell\_2,\|\cdot\|\_\infty)$ and the
identical map, the image is closed but the map isn't strict.
I don't know anything about normed spaces over non... | 2 | https://mathoverflow.net/users/21051 | 375323 | 156,606 |
https://mathoverflow.net/questions/375326 | 4 | Bartle-Graves theorem states that a surjective continuous linear operator between Banach spaces has a right continuous inverse that doesn't have to be linear.
Is it possible that this inverse is linear under additional assumption that an operator has a dense kernel?
| https://mathoverflow.net/users/167962 | Can the inverse operator in Bartle-Graves theorem be linear? | In general the right inverse of a linear bounded operator $T:X\to Y$ in Bartle-Graves theorem needs not to be linear, and there can be no linear bounded right inverse. In this case, any right inverse $S:Y\to X$ is everywhere not Fréchet differentiable nor Gateaux differentiable, because if $S$ is differentiable at $x\_... | 5 | https://mathoverflow.net/users/6101 | 375332 | 156,611 |
https://mathoverflow.net/questions/375318 | 3 | Let $\mathbb{G}= (A, \Delta)$ be a ($C^\*$-algebraic) compact quantum group. In a paper I'm reading, the space $A^\*= B(A, \mathbb{C})$ obtains a product
$$\omega\_1\*\omega\_2:= (\omega\_1\otimes \omega\_2) \circ \Delta$$
and this is used to prove the existence of the Haar functional on a compact quantum group.
*... | https://mathoverflow.net/users/nan | Convolution of functionals on compact quantum group | Every bounded linear functional on a $C^\ast$-algebra is a linear combination of states, so $\omega\_1\odot \omega\_2$ extends to the spatial (minimal) tensor product for all $\omega\_1, \omega\_2 \in A^\ast$ by a theorem of Takesaki.
| 4 | https://mathoverflow.net/users/126109 | 375337 | 156,614 |
https://mathoverflow.net/questions/375338 | 4 | Let $\mathbb{F}$ be an arbitrary field and consider a polynomial of **degree one**: $$f(x\_1,\ldots,x\_n)=a\_1x\_1+\cdots+a\_nx\_n+b\in \mathbb{F}[x\_1,\ldots,x\_n].$$
Let $H:f=0$ be the corresponding affine hyperplane and let $g\in\mathbb{F}[x\_1,\ldots,x\_n]$ be any polynomial that vanishes on $H$. If $\mathbb{F}$ is... | https://mathoverflow.net/users/33757 | Nullstellensatz for hyperplanes over a general field? | Applying a linear change of coordinates to the variables, we can assume without loss of generality that $f = x\_1$. Let $h(x\_2, \dots, x\_n) = g(0, x\_2, \dots, x\_n)$, considered as an element of $\mathbb{F}[x\_2, \dots, x\_n]$. Then $h(a\_2, \dots, a\_n) = 0$ for all $a\_2, \dots, a\_n \in \mathbb{F}$. This shows ho... | 6 | https://mathoverflow.net/users/31308 | 375342 | 156,615 |
https://mathoverflow.net/questions/374988 | 2 | $\DeclareMathOperator{\col}{\operatorname{col}}\DeclareMathOperator{\diag}{\operatorname{diag}}\DeclareMathOperator{\Range}{\operatorname{Range}}$Let $A \in \mathbb{R}^{n\times m}$, $D = \diag(d) = \diag (d\_1,...,d\_m)$ such that $d\_i \geq 0$ for all $i = 1,...,m$.
Consider the product $X = ADA^\top$. It is known t... | https://mathoverflow.net/users/156139 | Characterize matrix range | Since $\text{Range}(X)\subset\text{Range}(A)$, $$\text{Range}(X)=\text{Range}(A)\Leftrightarrow \text{rank}(X)=\text{rank}(A)\Leftrightarrow \ker(X)=\ker(A^T).$$
For any $M$ we have $MM^Tx=0\Leftrightarrow M^Tx=0$. Indeed $$0=\langle x,MM^Tx\rangle = \langle M^Tx,M^Tx\rangle =\|M^Tx\|^2.$$
Therefore $$x\in \ker(X)\Le... | 2 | https://mathoverflow.net/users/99045 | 375346 | 156,617 |
https://mathoverflow.net/questions/370081 | 7 | $\newcommand{\Mod}{\mathbf{Mod}} \newcommand{\map}{\mathrm{map}\_{E\_\infty-A}}$ Suppose $i:A\to B$ is a map of $E\_\infty$-ring spectra. It induces a functor of $\infty$-categories $\Mod\_B\to\Mod\_A$ by restriction of scalars.
A reasonable question is to ask when this is fully faithful; studying the counit of the r... | https://mathoverflow.net/users/102343 | Interesting "epimorphisms" of $E_\infty$-ring spectra | If $A$ is an $E\_\infty$ ring spectrum and $i : A \to B$ is any map of $A\_\infty = E\_1$ ring spectra such that the multiplication $\mu : B \wedge\_A B^{op} \to B$ is an equivalence, then $B \simeq LA$ where $L$ is some smashing Bousfield localization on the category of $A$-modules. In particular, $B$ will be $E\_\inf... | 2 | https://mathoverflow.net/users/9684 | 375353 | 156,620 |
https://mathoverflow.net/questions/375359 | 8 | This is kind of a strange and vague question... sorry about that.
I am really interested in $G\_2$ Twisted Connected sums as described in this paper: <https://arxiv.org/abs/math/0012189> "Twisted connected sums and special Riemannian holonomy" by Alexei Kovalev. I would like to use those constructions to come up with... | https://mathoverflow.net/users/nan | Are there mistakes in Kovalev's "Twisted connected sums and special Riemannian holonomy"? | The error in Kovalev's paper is described in [arXiv:1206.227](https://arxiv.org/abs/1206.2277) (see the discussion following theorem 2.6). An alternative proof is in [arXiv:1212.6929.](https://arxiv.org/abs/1212.6929)
>
> Building on the previous work of Tian–Yau, Kovalev claimed to prove
> the existence of exponen... | 12 | https://mathoverflow.net/users/11260 | 375360 | 156,623 |
https://mathoverflow.net/questions/375347 | 2 | I have an ideal $I$ generated by quadratic and cubic homogeneous polynomials in $10$ variables.
Macaulay2 tells me that $I$ defines an irreducible variety $X$ of dimension $5$ and degree $10$ in $\mathbb{P}^9$, and that $I$ is not radical.
When I ask for the primary decomposition of $I$ Macaulay2 gives me two ideal... | https://mathoverflow.net/users/nan | A question on a Macaulay2 computation | Commutative algebra is NOT the same as algebraic geometry, especially projective algebraic geometry.
The variety in $\mathbb{P}^9$ defined by $I$ and the variety in $\mathbb{P}^9$ defined by $I\_0$ are the same variety.
If you were to work in $\mathbb{A}^{10}$, then $I$ and $I\_0$ would define different affine sche... | 12 | https://mathoverflow.net/users/3077 | 375361 | 156,624 |
https://mathoverflow.net/questions/375272 | 2 | For how many $0 < d \leq D$ is there an integer solution to
$$x^2-dy^2 = -n$$
for $n > 1$? I have circumstantial reason to believe it might be $\sim D^{\frac{1}{2}}$ but I'd be interested in any upper bound that is $o(D)$.
| https://mathoverflow.net/users/145167 | Density of $d$ for which a generalized Pell equation has a solution | To expand on Dan's answer, for general $n \in \mathbb{Z}$ the equation
$$\displaystyle x^2 - dy^2 = n$$
is soluble in integers $x,y$ only if the quadratic form
$$\displaystyle q\_{d,n}(x,y,z) = x^2 - dy^2 - nz^2$$
is everywhere locally soluble, i.e., it is soluble over $\mathbb{R}$ and over $\mathbb{Q}\_p$ for ... | 3 | https://mathoverflow.net/users/10898 | 375362 | 156,625 |
https://mathoverflow.net/questions/374267 | 7 | Given a segment and a value $c$ less than the segment length, let $A\_1,\dots,A\_n$ be finite unions of intervals on the segment. We choose a finite union of intervals $B$ with $|B|=c$ that maximizes $|B\cap A\_1|\times\dots\times |B\cap A\_n|$, where $|\cdot|$ denotes the length (i.e. Lebesgue measure). If there are m... | https://mathoverflow.net/users/83212 | Shrinking subset and product | OK, it seems that this is a counterexample.
Take 8 disjoint segments $I\_1,\dots,I\_8$ of length 1. Take 8 sets
$$
S\_1=\{1,2,3\}, \quad
S\_1=\{4,5,6\}, \quad
S\_1=\{1,2,4\}, \quad
S\_1=\{1,2,5\}, \quad
S\_1=\{1,2,6\}, \quad
S\_1=\{1,3,4\}, \quad
S\_1=\{1,3,5\}, \quad
S\_1=\{1,3,6\}.
$$
Say that $I\_i$ lies i... | 2 | https://mathoverflow.net/users/17581 | 375367 | 156,626 |
https://mathoverflow.net/questions/375040 | 0 | If $\nu$ is a finite measure on $(\mathbb R,\mathcal B(\mathbb R))$, let $\nu^{\ast k}$ denote the $k$-fold convolution¹ of $\nu$ with itself for $k\in\mathbb N\_0$, $$\exp(\nu)\mathrel{:=}\sum\_{k=0}^\infty\frac{\nu^{\ast k}}{k!}$$ and $$\operatorname{CPoi}\_\nu\mathrel{:=}\frac{\exp(\nu)}{\exp(\nu)(\mathbb R)}.$$ Mor... | https://mathoverflow.net/users/91890 | If $\mu$ is an infinitely divisible probability measure on $[0,\infty)$, then the Lévy measure of $\mu$ is the vague limit of $n\mu^{*1/n}$ | There are at least three ways to show that $n \mu^{\*1/n}$ converges to $\nu$ vaguely in $\mathbb{R} \setminus \{0\}$. Let $X\_t$ be the Lévy process such that $X\_1$ has distribution $\mu$, and let $f$ be a smooth, compactly supported $f$ on $\mathbb{R} \setminus \{0\}$. It is sufficient to show that $t^{-1} \mathbb{E... | 1 | https://mathoverflow.net/users/108637 | 375372 | 156,628 |
https://mathoverflow.net/questions/375370 | 10 | I would like an example showing that one of the most basic induction approaches to the union-closed conjecture fails. If, for any union-closed family $\mathcal{A}$ of subsets of a finite set $X$, there is some $x \in X$ such that each $y \in X$ has $|\{A \in \mathcal{A} : A \ni y \text{ and } A \ni x\}| \ge \frac{1}{2}... | https://mathoverflow.net/users/129185 | Union closed conjecture induction | Let $X = 123456$, and $\mathcal{A} = \{134, 1345, 1346, 13456, 256, 2356, 2456, 23456, 123456\}$. Let $f\_x$ and $f\_{x, y}$ be the number of sets in $\mathcal{A}$ containing $x$, or both $x$ and $y$ respectively. Say that $y$ is rare for $x$ if $f\_{x, y} < f\_x / 2$. Then:
* $f\_1 = f\_2 = 5$, but $f\_{1, 2} = 1$, ... | 11 | https://mathoverflow.net/users/106512 | 375373 | 156,629 |
https://mathoverflow.net/questions/374222 | 1 | I was just looking through some of my old questions on StackExchange and noticed that this one went totally unresolved. I still think it'd be useful as a lemma for plane topology if it were true, and would be curious to see a counterexample if not.
Suppose we are given two pairs $(U, V)$ and $(Y, Z)$ of connected, si... | https://mathoverflow.net/users/110965 | Isotopy Classes of Non-Connected Planar Sets | Neither additional condition 1, nor condition 2 is sufficient for $A≃B$.
A small modification of your example can witness this:
Let $U=Y$ be the 'thickened open topologist's sine curve' on the interval $(0,π)$ limiting to $S=\{0\}\times [-1,1]$ and let $V$ be the reflection of $U$ with respect to the $y$-axis.
Let $Z$ ... | 2 | https://mathoverflow.net/users/128723 | 375388 | 156,632 |
https://mathoverflow.net/questions/372349 | 3 | $\DeclareMathOperator{\Q}{\mathbb{Q}}$Call "L-rig" any class $\mathcal{L}$ of L-functions of automorphic representations of $\operatorname{GL}\_{n}(\mathbb{A}\_{\Q})$ for some $n$ belonging to the Selberg class that be closed under both the usual product (which we'll denote by $\times$) and the Rankin-Selberg convoluti... | https://mathoverflow.net/users/13625 | Are there infinitely many L-rigs? | The Rankin-Selberg convolution of a quadratic Dirichlet L-function with itself is the Riemann zeta function. Therefore the rig generated by $\{1, \zeta(s), L(s, \chi\_d)\}$ consists of all finite products (and powers) of $\zeta(s)$ and $L(s, \chi\_d)$. In particular, there are infinitely many L-rigs.
If you start wit... | 7 | https://mathoverflow.net/users/10220 | 375393 | 156,634 |
https://mathoverflow.net/questions/375374 | 7 | Mathematicians work a lot and are usually inspired by many things. In their lifetimes they get to publish only portions of their results. There have been stories of how Gauss, Euler, Ramanujan, Einstein, Hilbert etc kept notebooks and in it we can find glimpses to their thought processes and glimpses to results they di... | https://mathoverflow.net/users/10035 | Do mathematicians use notebooks to keep their results these days? | 1. Yes, mathematicians keep notebooks. Sometimes, after their death, notebooks are published.
Here is an example: <http://www.claymath.org/publications/quillen-notebooks>
Here is another example, though not so recent: <https://www.math.uu.se/collaboration/beurling/unpublished-manuscripts/>
I believe many mathematic... | 5 | https://mathoverflow.net/users/25510 | 375396 | 156,636 |
https://mathoverflow.net/questions/375397 | 5 | I've already asked the question on [MSE](https://math.stackexchange.com/questions/3703824/simple-condition-that-would-prove-a-function-transcendental) but there are still no answers, so I'm going to ask it here.
I conjectured that for every algebraic function $f(x)$ that is differentiable on $\mathbb{R}$, its $\lim\_... | https://mathoverflow.net/users/168000 | "Simple" condition that would prove a function transcendental | $\newcommand\R{\mathbb R}$Your conjecture is true. Indeed, suppose that a function $f\colon\R\to\R$ is continuous and algebraic, so that
$$\sum\_{j\in[n]\_0}p\_j(x)f(x)^j=0\tag{1}$$
for all $x\in\R$, where $[n]\_0:=\{0,\dots,n\}$, $n$ is a natural number, and, for each $j\in[n]\_0$, $p\_j$ is a polynomial function of s... | 4 | https://mathoverflow.net/users/36721 | 375400 | 156,637 |
https://mathoverflow.net/questions/375402 | 4 | I read through the celebrated paper of Cohen - Lenstra heuristics. But unfortunately, the Cohen - Martinet paper is originally written in French, which I do not understand. So I would like to know if there are any English references for this paper.
Also, may I know in a very short summary what extensions have been ad... | https://mathoverflow.net/users/167999 | English references to Cohen - Martinet Heuristics | The ultimate goal of the heuristics I made with Martinet was to generalize
the C-L heuristics to an arbitrary extension $L/K$ of number fields, first
assuming $L/K$ galois, then more generally. Although the basic ideas were
sound, many modifications needed to be made since the original publication.
First, we had to dec... | 6 | https://mathoverflow.net/users/81776 | 375411 | 156,638 |
https://mathoverflow.net/questions/375368 | 3 | I have a general question about the motivation behind to definition the smooth morphisms
as we know it from algebraic geometry. The most common
definition of a smooth morphism $: X \to Y$ between two smooth Noetherian schemes
$X,Y$ is:
$f$ is smooth if and only if
(i) $f$ is flat and locally of finite presentation
... | https://mathoverflow.net/users/108274 | Smooth morphism (algebraic geometry) vs. Submersion (differential geo) & Ehresman's Lemma | One of the many equivalent definitions of smoothness of a morphism $f\colon X\to Y$ of varieties over a field $k$ is that $f$ is smooth if and only if it is *formally smooth*. The latter means the following: given any square-zero extension of $k$-algebras $S\to R$ and a commuting square
$\require{AMScd}$
\begin{CD}
\ma... | 12 | https://mathoverflow.net/users/126183 | 375415 | 156,641 |
https://mathoverflow.net/questions/375375 | 2 | Take a line $L$ of the first type on a smooth cubic threefold $X$ over $\mathbb C$, then its normal bundle $N\_{L|X}$ is isomorphic to $\mathcal{O}\_L\oplus \mathcal{O}\_L$. This is equivalent to say that there is a $\mathbb P^1$-family of quadric surfaces in $\mathbb P^4$ tangent to $X$ along $L$. I'm trying to write ... | https://mathoverflow.net/users/74322 | Quadric surfaces tangent to a cubic threefold along a line of first type | It turns out that my question is all about linear algebra. As I mentioned in the chat, to find such a quadric surface, one needs to find a line $L'$ in $\mathbb P^4$, which lies in the normal direction of $L$ inside the cubic threefold $X$. Such a line is the image of a nonzero section $s$ in the normal bundle $N\_{L|X... | 0 | https://mathoverflow.net/users/74322 | 375422 | 156,643 |
https://mathoverflow.net/questions/375427 | 0 | Consider simple graphs on the same vertex set $[n]$. For two graphs $G, H$, let $d(G, H) = \min\_{H' \sim H} |E(G) \triangle E(H')|$ — the smallest number of edge additions/removals needed to make $G$ isomorphic to $H$.
Let $D(G) = \max\_H d(G, H)$, and $S(G) = \max(|E(G)|, {n \choose 2} - |E(G)|)$. Clearly, $D(G) \g... | https://mathoverflow.net/users/106512 | Making graphs isomorphic with edge additions/removals | Yes. Suppose $G$ has density $d$ less than 1/2. Randomly reorder the vertices of $H$ and make the necessary edits to $G$ to get this particular copy of $H$.
Suppose $H$ has density $p$. The probability of editing a given edge of $G$ is $1-p$, and of editing a non-edge is $p$. So by linearity of expectation we edit a ... | 1 | https://mathoverflow.net/users/36212 | 375428 | 156,644 |
https://mathoverflow.net/questions/375426 | 9 | Suppose we have a (say compactly supported) $C^0$-vector field $X:\mathbb R^n\to\mathbb R^n$ such that for every $x\in\mathbb R^n$ there is a unique $C^1$-curve $\gamma:\mathbb R\to\mathbb R^n$ solving $\dot\gamma\_x(t)=X(\gamma\_x(t))$ with $\gamma\_x(0)=x$.
Then the ode flow $\mathcal F\_X$ is pointwisely defined i... | https://mathoverflow.net/users/118469 | Does ODE uniqueness unconditionally implies the flow continuity? | Yes, the uniqueness of solutions implies continuous dependence on initial conditions and parameters. See Theorem 3.2 in Hartman's "Ordinary differential equations".
| 6 | https://mathoverflow.net/users/85336 | 375437 | 156,646 |
https://mathoverflow.net/questions/375448 | 2 | Suppose a random variable $X$ is distributed as $\operatorname{NB}(\mu, \theta)$, and its mass is as follows
$$ \mathrm{P}(X = y) = \binom{y + \theta - 1}{y} \left(\frac{\mu}{\mu + \theta}\right)^{y}\left(\frac{\theta}{\mu + \theta}\right)^{\theta}.$$
Does anyone know how to calculate the expectation of $1 / X$ in this... | https://mathoverflow.net/users/153595 | Finding the expectation $\mathrm{E} (1/ X)$ for a negative binomial random variable $X$ | Mathematica answers your second question for concrete values of $n$ (e.g. $n=3$) by
```
Mean[TransformedDistribution[X/(c + X)^3, X \[Distributed] NegativeBinomialDistribution[\[Mu], \[Theta]],Assumptions->c>0]]
```
$$-\frac{(\theta -1) \mu \theta ^{\mu } \, \_4F\_3(c+1,c+1,c+1,\mu +1;c+2,c+2,c+2;1-\theta )}{(c+1)... | 1 | https://mathoverflow.net/users/35959 | 375454 | 156,652 |
https://mathoverflow.net/questions/375447 | 9 | Does every open orientable even-dimensional smooth manifold admit an almost complex structure?
| https://mathoverflow.net/users/164384 | Does every open orientable even-dimensional smooth manifold admit an almost complex structure? | If $M$ admits an almost complex structure, then the odd Stiefel-Whitney classes vanish and the even Stiefel-Whitney classes admit integral lifts, namely $c\_i(M) \equiv w\_{2i}(M) \bmod 2$. These two conditions give restrictions on the smooth manifolds which can admit almost complex structures.
The first restriction,... | 24 | https://mathoverflow.net/users/21564 | 375457 | 156,653 |
https://mathoverflow.net/questions/375410 | 1 | A matrix, $\mathbf{A}(\theta)\in\mathbb{R}^{n\times n}$, has elements that depend on a parameter $\theta$. The $j$-th eigenvalues and eigenvectors of the matrix are denoted as $\lambda\_j$ and $\mathbf{x}\_j$, respectively.
I would like to know the requirements to obtain finite backward derivatives of eigenvectors in d... | https://mathoverflow.net/users/151615 | Requirements for finite backward derivatives of degenerate eigenvectors | Let's take $m \in \mathrm{degen}(n)$. The terms involving $m$ and $n$ in $\frac{\partial \mathcal{L}}{\partial \mathbf{A}}$ are
$$
\frac{\partial\mathcal{L}}{\partial \mathbf{A}} =-(\lambda\_m -\lambda\_n)^{-1}\left[\mathbf{x}\_m\mathbf{x}\_m^T\frac{\partial\mathcal{L}}{\partial \mathbf{x}\_n}\mathbf{x}\_n^T - \mathbf{... | 0 | https://mathoverflow.net/users/151615 | 375481 | 156,662 |
https://mathoverflow.net/questions/374793 | 7 | Let $G$ be a graph s.t. any two cycles $C\_1, C\_2 \subseteq G$ either have a common vertex or $G$ has an edge joining a vertex in $C\_1$ to a vertex of $C\_2$. Equivalently: for every cycle $C$ the graph obtained from $G$ by deleting $C$ and all neighbors of $C$ is acyclic. Let's denote the class of all such graphs by... | https://mathoverflow.net/users/37432 | Tree-width of graphs in which any two cycles touch | I tried to prove the statement for a while and I think I managed to narrow it down to one particularly difficult case. In the end, it led me to a counter example, showing there are no such values $g$ and $t$. This came as a bit of a surprise for me. The construction goes as follows.
(1) For every $n \geq 1$ there is ... | 3 | https://mathoverflow.net/users/37432 | 375488 | 156,667 |
https://mathoverflow.net/questions/371858 | 1 | I need to cover a case of $n$-dimensional locally symmetric Riemannian space of rank $n-1$. Is there a simple proof that there is no such irreducible space ($n>4$)? If I need to cite the Cartan classification for that, which concrete paper/book you suggest?
| https://mathoverflow.net/users/51043 | Riemannian symmetric space of dimension $n$ and rank $n-1$ | An irreducible symmetric space $M$ of dimension $n$ and rank $n-1$ has dimension $n=2$.
Indeed, the rank is the codimension of a principal orbit of the isotropy group $K$ at $p \in M$ acting on the tangent space $T\_pM$. So the $K$-orbits are one dimensional. Now any monoparametric subgroup of $K$ has a one dimensional... | 2 | https://mathoverflow.net/users/43122 | 375490 | 156,668 |
https://mathoverflow.net/questions/375492 | 0 | I asked the following question on mathstack but didn't receive any answers. I suspect that this question has a simple answer but I haven't thought about Fourier transforms in a while so am being sluggish at figuring it out. Any help/comments/suggestions would be appreciated.
Consider a function $f$ that has smooth Fo... | https://mathoverflow.net/users/80930 | Integration against a certain Fourier transform | By Fourier inversion formula,
$$ \int\_{-\infty}^\infty e^{-2\pi i u x} \hat f(u) du = f(x) $$
for all $x$ (up to some constant factors that depend on normalisation of the Fourier transform). By Fubini's theorem,
$$ \int\_0^1 e^{h x} f(x) dx = \int\_{-\infty}^\infty \frac{e^{h - 2\pi i u} - 1}{h - 2 \pi i u} \hat f(u) ... | 4 | https://mathoverflow.net/users/108637 | 375498 | 156,671 |
https://mathoverflow.net/questions/375472 | 7 | The regulator of a number field is essentially the covolume of the unit group embedded into the vector space $\{(x\_1, \ldots, x\_{r+s}): \sum\_i x\_i=0\}$ under the log embedding: $$x \mapsto (\log |\sigma\_1(x)|, \ldots \log |\sigma\_r(x)|, \log |\sigma\_{r+1}(x)|^2, \ldots , \log |\sigma\_{r+s}(x)|^2).$$
But you n... | https://mathoverflow.net/users/22 | Why is the regulator of a number field normalized the way it is? | Let $V$ be a vector space and $W$ a subspace.
Given a volume on $V$ and a volume on $V/W$, we can define a volume on $W$. (Here "volume" = "translation-invariant measure" but no real measure theory is being used. We could also define a volume as an element of the top wedge power of the dual vector space.)
Geometric... | 4 | https://mathoverflow.net/users/18060 | 375499 | 156,672 |
https://mathoverflow.net/questions/375450 | 1 | In my research I encounter some elements in a root lattice and I would like to verify that these elements are imaginary roots. Consider the root system $J\_{6, 11}$ with the following Dynkin diagram:
\begin{align}
\circ - \circ - \circ - \circ - \circ - & \circ - \circ - \circ - \circ - \circ \\
& \ | \\
& \ \bullet
\e... | https://mathoverflow.net/users/11877 | How to verify that an element in the root lattice is an imaginary root of a non-hyperbolic root system? | Denote
$$K = \{ \alpha\in Q\_+\setminus\{0\} \mid \langle \alpha,\alpha\_i^\vee \rangle \leqslant 0 \text{ for all $i$ and $\operatorname{supp}(\alpha)$ is connected} \}.$$
Here $Q\_+$ is the positive part of the root lattice and $\operatorname{supp}(\alpha)$ is the support of $\alpha$, that is, the subdiagram of the D... | 1 | https://mathoverflow.net/users/5018 | 375500 | 156,673 |
https://mathoverflow.net/questions/375462 | 7 | I am having a hard time finding examples of computations of normal invariants of surgery theory (or more generally the set of homotopy classes of maps $[X,G/O]$). Does anybody have good references?
| https://mathoverflow.net/users/147200 | Normal invariants | Many examples of computations of $[M,G/O]$ appear in papers which apply surgery theory. Here are some examples:
* [Brumfiel](https://www.maths.ed.ac.uk/%7Ev1ranick/papers/brumfiel.pdf) did the complex projective spaces $\mathbb{C}P^n$.
* [Land](http://markus-land.de/app/download/2771195/Normal+invariants.pdf) did the... | 5 | https://mathoverflow.net/users/798 | 375502 | 156,675 |
https://mathoverflow.net/questions/375494 | 0 | Let $a\_1, a\_2, \ldots a\_n$ and $b\_1, b\_2, \ldots b\_n$ be two sequences of $n\gg 1$ real numbers such that, for all $1\le i\le n$, we have $0<a\_i \le b\_i\le 1$. Let the ratio $R$ defined as follows:
$$R:=\frac{\sum\_{1\le i\le n} a\_i}{\sum\_{1\le i\le n} b\_i }~.$$
---
**Question:** How can we set value... | https://mathoverflow.net/users/115803 | Bounding the ratio of the $\ell_1$-norms of two real-valued $n$-vectors as a linear combination of their $n$ element-wise ratios | For each $i$, let
$x\_i:=r\_\*/(nr\_i)$, where $r\_\*:=\min\_j r\_j$, so that $x\_1,\dots,x\_n$ are functions of the $r\_i$'s. Then
$$R'=\sum\_{i=1}^n x\_i r\_i=r\_\*\le R,\tag{1}$$
so that $R'$ is a lower bound on $R$. This bound is tight, since $R'=r\_\*=R$ if the $r\_i$'s are the same for all $i$.
The inequality i... | 1 | https://mathoverflow.net/users/36721 | 375503 | 156,676 |
https://mathoverflow.net/questions/375504 | 18 | Let $M$ be a connected finite dimensional topological manifold and $g$ be any metric on it that induces the topology of $M$ ($g$ is not a Riemannian metric). How to prove that the group of isometries of $(M,g)$ is a finite dimensional Lie group?
**PS.** Note that in case $M$ is a Riemannian manifold, this statement i... | https://mathoverflow.net/users/13441 | The group of isometries of a manifold is a Lie group, isn't it? | The assertion (for connected topological manifolds) is equivalent to the Hilbert–Smith conjecture. The latter (in one of its equivalent versions) asserts that every continuous action of every profinite group on every topological metrizable manifold has an open kernel.
Indeed, let $G$ be a profinite group acting conti... | 15 | https://mathoverflow.net/users/14094 | 375509 | 156,679 |
https://mathoverflow.net/questions/375489 | 7 | I`m reading Melvin B. Nathanson's "Elementary Methods in Number Theory"
and I can't think of a way of deducing Selberg's formula (9.3) from the prime number theorem.
This is one of the tasks left for the reader at the end of chapter 9.3 (The Elementary Proof).
$$(9.3) \qquad \qquad \vartheta(x) \log x + \sum\_{p \leq... | https://mathoverflow.net/users/168078 | How to use the Prime Number Theorem in order to prove Selberg's Formula? | The $O(x)$ term can be refined by completely elementary means that are not at all complicated. Filip Saidak (On the prime number lemma of Selberg, Math. Scand. 103 (2008), no. 1, 5–10) proved a much clearer connection between Selberg's result and the prime number theorem. He stated this for $\psi(x) = \sum\_{n\leq x}\L... | 6 | https://mathoverflow.net/users/111215 | 375513 | 156,681 |
https://mathoverflow.net/questions/375176 | 0 | I have the equation:
$$
\dot{x}\_i = F\_i(x)
\tag{1}
$$
with $x\in \mathbb{R}^n$. To deal with the Lyapunov exponents, we write the equation for small displacements $\delta x\_i$:
$$
\dot{\delta x}\_i = \sum\_j \frac{\partial}{\partial x\_j} F\_i(x) \delta x\_j
\tag{2}
$$
The rate of increase of the vectors is related ... | https://mathoverflow.net/users/138060 | Lyapunov vectors along a trajectory | The confusion indeed concerns the order of $Y$ and $Y^\*$ (I prefer to use $\*$ instead of $T$ for transposition) in the definition of the matrix $M$. This is quite common, and the reason is that both orders actually do occur - depending on how the increments are added in the definition of the matrices $Y(t)$. Let me f... | 1 | https://mathoverflow.net/users/8588 | 375514 | 156,682 |
https://mathoverflow.net/questions/375515 | 0 | I am reading the paper [Barycenters in the Wasserstein space](https://doi.org/10.1137/100805741) by Martial Agueh and Guillaume Carlier. In Proposition 3.5, they prove the existence and uniqueness of
$$\nu \mapsto \sum\_{i=1}^p \frac{\lambda\_i}{2}W\_2^2(\nu\_i, \nu).$$
There is no mention of uniqueness in the proof.... | https://mathoverflow.net/users/168083 | The uniqueness of Barycenters in the Wasserstein space | Uniqueness follows directly from proposition 3.3 on the previous page. For completeness, a shortened version containing the relevant parts:
>
> Suppose $\mu$ and $\nu$ are probability measures on $\mathbb{R}^d$ with finite second moment and $\mu$ does not give mass to small sets. There exists a $\mu$-almost surely ... | 3 | https://mathoverflow.net/users/165966 | 375530 | 156,687 |
https://mathoverflow.net/questions/375543 | 2 | Let $\Omega\subset\mathbb{R}^n$ open, bounded with smooth boundary, let $s\in(0,1)$. I know that the fractional Laplacian has a sequence of eigenfunctions $\{e\_k\}\_{k\in\mathbb{N}}\subset H^s(\mathbb{R}^N)$, $e\_k=0$ a.e. on $\mathbb{R}^n\setminus\Omega$, $\forall k\in\mathbb{N}$. Moreover I know that these eigenfunc... | https://mathoverflow.net/users/167027 | Eigenfunctions of the fractional Laplacian are smooth? | This is consequence of the theory of Pseudo-differential operators. Your fractional Laplacian $P\_s$ is a PDO of order $2s$. Above all, it is elliptic. This is a classical result (certainly in Hörmander) that if $P\_su\in H^r\_{\rm loc}$, then $u\in H^{r+2s}\_{\rm loc}$, where $H^{\cdots}$ is the class of Sobolev space... | 4 | https://mathoverflow.net/users/8799 | 375545 | 156,688 |
https://mathoverflow.net/questions/375538 | 10 | In categorical set theory, we observe that certain topoi satisfy (suitable versions of) certain axioms from set theory. For example, Lawvere's $\mathsf{ETCS}$ asserts that $\mathbf{Set}$ is a well-pointed topos with a natural numbers object, satisfying the (internal) axiom of choice. $\mathsf{ETCS}$ is known to be equi... | https://mathoverflow.net/users/136473 | When does a topos satisfy the axiom of regularity? | The relationship between toposes and set theories was studied comprehensively in
>
> Steve Awodey, Carsten Butz, Alex Simpson, Thomas Streicher: [Relating first-order set theories, toposes and categories of classes](https://www.sciencedirect.com/science/article/pii/S0168007213000730).
> Annals of Pure and Applied L... | 21 | https://mathoverflow.net/users/1176 | 375548 | 156,690 |
https://mathoverflow.net/questions/375531 | 6 | Let $\Gamma$ be a countable (discrete) group and let $\varphi:\Gamma\times\Gamma\to\mathbb{C}$ be a (non-equivariant) Schur multiplier. See Chapter 5 of [2] for details. Assume that, for all $t\in\Gamma$, the function
\begin{align\*}
s\longmapsto\varphi(st,s)
\end{align\*}
is weakly almost periodic. Let $m$ be the uniq... | https://mathoverflow.net/users/168082 | Averaging weakly almost periodic Schur multipliers | Ignacio, the answer is yes. Indeed, there is a net $m\_i$ of probability measures on $G$ such that $m(f) = \lim\_i \int f dm\_i$ for every $f \in \textrm{WAP}(\Gamma)$. One justification of this is as follows: extend $m$ to a (perhaps non-invariant) mean on $\ell\_\infty(\Gamma)$, and use the standard weak-\* density o... | 6 | https://mathoverflow.net/users/10265 | 375551 | 156,691 |
https://mathoverflow.net/questions/375549 | 0 | If $A$ is a subset of the set of positive integers $\mathbb{N}$, there are (at least) two notions of what it means for $A$ to be *thin*:
1. $A$ is thin in the 1st sense if $\lim\sup\_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n} = 0$; and
2. $A$ is thin in the 2nd sense if $\sum\_{a\in A}\frac {1}{a} < \infty$.
As us... | https://mathoverflow.net/users/8628 | Two notions of thinness of subsets of $\mathbb{N}$ | Since $|A \cap \{1,\ldots,n\}| / n \leq \sum\_{a \in A} \frac{1}{a}$ 2. implies 1. But if $A$ is the set of primes 1. holds true and 2. is false.
| 4 | https://mathoverflow.net/users/100904 | 375553 | 156,692 |
https://mathoverflow.net/questions/375539 | 3 | let $X,Y$ be smooth schemes (or rigid spaces etc..) over a base $S$, let $f:Y \rightarrow X$ be a $S$-morphisn and let $\mathcal{F}$ be a locally free $\mathcal{O}\_X$-module with connection $\nabla$. How do we define the "pull-back connection" along $f$? We get a connection $$f^{-1}\mathcal{F} \overset{f^{-1}\nabla}{\... | https://mathoverflow.net/users/168099 | Pullback of a connection | Recall that you have a map $f^\* \colon f^{-1} \Omega\_X \to \Omega\_Y$ (pull-back of differentials). Consider the composition $$ \nabla' \colon f^{-1} E \xrightarrow{f^{-1} \nabla} f^{-1} E\otimes\_{f^{-1} \mathcal{O}\_X} f^{-1}\Omega\_X \xrightarrow{{\rm id}\otimes f^\*} f^{-1} E\otimes\_{f^{-1}\mathcal{O}\_X} \Omega... | 4 | https://mathoverflow.net/users/3847 | 375559 | 156,693 |
https://mathoverflow.net/questions/375554 | 20 | I am reading a paper on stochastic optimization. And in this paper, the proofs are based on the Pinelis' 1994 inequality. I read the paper by Pinelis for more information and it is with great frustration that I was not able to find the inequality corresponding to that mentioned in the paper I am reading.
Here's the i... | https://mathoverflow.net/users/153055 | Does Pinelis' inequality (1994) exist? | As noted in Carlo Beenakker's comment, your inequality is a direct application of Theorem 3.5 in the linked paper: in that theorem, take $d\_j=X\_j$, $r=\sqrt{2M^2 T\ln(2/\delta)}$, $b\_\*^2=M^2T$, and $D=1$.
| 82 | https://mathoverflow.net/users/36721 | 375560 | 156,694 |
https://mathoverflow.net/questions/375558 | 11 | Let natural numbers $a, b > 2$ be adjacent if $|a^2 - b^2|$ is a square number. One can find a 3-clique.
For example 153, 185, 697. The questions are: does there exist a 4-clique? Is this graph connected?
| https://mathoverflow.net/users/167798 | 4-cliques of pythagorean triples graph and its connectivity | Seems that here is a proof that the graph is connected
Let $C$ be a component.
1. $C$ contains an even number, as an odd number $2k+1$ is adjacent to $2k(k+1)$.
2. $C$ contains a number divisible by $6$: if $2k\in C$ with $3\nmid k$, then $k^2-1\in C$ as well.
3. If $3k\in C$ and $p$ is a prime, then $3kp\in C$. In... | 9 | https://mathoverflow.net/users/17581 | 375574 | 156,699 |
https://mathoverflow.net/questions/375578 | 4 | Let $V$ be a finite dimensional complex vector space. Let $G$ be a compact group with normalized Haar measure $\mu$. In the representation theory of compact groups, I encounter
$$\int\_G f(g) \mu(dg)$$
where $f: G \to V$ is a continuous function.
How is this integral definied? I know about the Lebesgue integral f... | https://mathoverflow.net/users/nan | Integration in a finite dimensional vector space | If you do not want to use coordinate systems, a more intrinsic way to reduce to scalar functions is defining $\int\_Gf(g) \mu(dg)$ as the (unique) element $v\in V$ such that for any linear form $\phi\in V^\*$ one has $\int\_G \langle \phi,f(g)\rangle\mu(dg)=\langle \phi,v\rangle$.
| 6 | https://mathoverflow.net/users/6101 | 375579 | 156,700 |
https://mathoverflow.net/questions/375577 | 0 | For each $ m \ge 1$, let $X\_m$ and $Y\_m$ be two **non-negative** **iid** random variables with the same distribution. (The distributions of $X\_m$ may change with different $m$.)
\*\*Assume that their support of $X\_m$ is $[0, \infty),$ so we're not considering point mass situation.
Denote by $\to\_{p}$ the conve... | https://mathoverflow.net/users/35936 | Question on limit in probability of the ratio of max to min of 2 sequences of non-ive, continuous iid random variables with support $[0, \infty).$ | The answer is no, and the additional support/non-discreteness condition is of no help.
Consider e.g. the following modification of the counterexample in Matt F.'s comment. Suppose that $P(U\_m=0)=1/m=P(U\_m=m^2)$ and $P(U\_m=1)=1-2/m$ for $m\ge2$. Let $V\_m$ be independent of $U\_m$ and have the exponential distribut... | 3 | https://mathoverflow.net/users/36721 | 375581 | 156,701 |
https://mathoverflow.net/questions/375606 | 0 | We are given a set $S=\{1, 2, \ldots, n\}$ where $n\gg 1$, and for all indices $1\le i \le n$, $i$ is associated with a real value $\alpha\_i\!\cdot\! v\_i$, where $\alpha\_i\in[0,1]$ and $v\_i\in(0,1]$.
Let $X$ be a discrete random variable whose sample space consists of $n'<n$ (not necessarily disjoint) proper subs... | https://mathoverflow.net/users/115803 | Expectation of the ratio of two discrete random variables with combinatorial constraints | This obviously depends on the "closeness" measure of the lower bound. FWIW, for any $n$ (let $n = 2k$ even for simplicity) and the following input values:
* $\alpha\_1 = \alpha\_k = 0$, $\alpha\_{k + 1} = \alpha\_{2k} = 1$,
* $v\_1 = v\_2 = \ldots = v\_{2k} = 1$,
* $p(1) = \ldots = p(2k) = \frac{1}{k + 1}$,
$\mathb... | 1 | https://mathoverflow.net/users/106512 | 375617 | 156,712 |
https://mathoverflow.net/questions/375188 | 6 | Each function $f\in\mathbb{C}(t)$ can be rewritten in the form $f = a\_{k}t^{k}+\ldots+a\_0+a\_1t+\ldots$, $k\in\mathbb{Z}$ and it is possible to define the topology with the open prebase at zero
$V\_{n,v,\varepsilon} = \{f=a\_kt^k+\ldots+a\_0+a\_1t+\ldots\in \mathbb{C}(t)| \sum\_{k\leq m<n}|m|^v|a\_m|<\varepsilon\}$... | https://mathoverflow.net/users/100140 | What is good $t$-adic like topology on $\mathbb{C}(t)$? | Let $\|f\|\_{n, m} = \sum\_{i<n}(|i|^m+10)|a\_i|$ denote a family of seminorms on $\mathbb{C}(t)$. Note that for any $\{(n\_v, m\_v)\}\_{v\text{ in finite subset of }\mathbb{N}}$ there is a seminorm $\|.\|\_{n,m}$ such that $\|f\|\_{n, m}\geq\sup\_v\|f\|\_{n\_v, m\_v}$. Also these seminorms define the same topology as ... | 0 | https://mathoverflow.net/users/100140 | 375623 | 156,715 |
https://mathoverflow.net/questions/375626 | 3 | Let I have the following function,
$f = Q(a\Re (x + y))Q(b\Im (x + y))\log \left\{ {Q(a\Re (x + y))Q(b\Im (x + y))} \right\}$
Where, $x,y \in C$, $a,b\in R$ and $- m \le \Re (x),\Re (y),\Im (x),\Im (y) \le m$, $m$ is a finite real number.
And $Q\left( z \right) = \frac{1}{{\sqrt {2\pi } }}\int\limits\_z^\infty {{... | https://mathoverflow.net/users/160736 | Analyticity of $f = Q(a\Re (x + y))Q(b\Im (x + y))\log \left\{ {Q(a\Re (x + y))Q(b\Im (x + y))} \right\}$ in the complex plane? | The answer is: of course not. Indeed, take any real $a\ne0$, any real $b$, and any real $m>0$. If $f$ were analytic in $x,y$ such that $-m\le\Re x,\Re y,\Im x,\Im y\le m$ then it would be analytic in $x$ at $x=0$ for $y=0$. Then the real and imaginary parts of the function
$$\mathbb R^2\ni(s,t)\mapsto g(s,t):=Q(as)Q(bt... | 3 | https://mathoverflow.net/users/36721 | 375640 | 156,723 |
https://mathoverflow.net/questions/375641 | 0 | Let $\mathcal{P}\_2(\mathbb{R}^n)$ be the set of Borel probability measures on $\mathbb{R}^n$ with finite mean and variance; in the sense that
$$
\int\_{x \in \mathbb{R}^n} \|x\|^p d\mathbb{P}(x) < \infty,
$$
for $p=1,2$. Let $\mathcal{G}(\mathbb{R}^n)\subseteq \mathcal{P}(\mathbb{R}^n)$ be the set of Gaussian measures... | https://mathoverflow.net/users/36886 | Projection onto manifold of Gaussian measures by "trunction" of moments | Such a metric is given by
$$d(P,Q):=2\times1\{(\mu\_P,\Sigma\_P)\ne(\mu\_Q,\Sigma\_Q)\}+\sup\_B|P(B)-Q(B)|$$
for Borel probability measures $P$ and $Q$ on $\mathbb{R}^n$ with finite second moments, where the $\sup$ is taken over all Borel subsets $B$ of $\mathbb{R}^n$.
| 1 | https://mathoverflow.net/users/36721 | 375642 | 156,724 |
https://mathoverflow.net/questions/375613 | 9 | I am trying to understand why (at least my most elementary understanding of) topological K-theory breaks down for non-compact things (which I have seen asserted in various places). In particular, as someone who mainly thinks about manifolds, I would like to understand what fails on non-compact manifolds. But let's be a... | https://mathoverflow.net/users/147463 | K-theory on finite-dimensional (possibly not finite) CW complexes | I am posting an answer to my own question, so that it will be marked as answered and may possibly resolve a similar confusion for someone else, in the future. This question was of the form "People keep saying that something goes wrong and I can't figure out what!" So the answer of "Nothing goes wrong" from Tom Goodwill... | 4 | https://mathoverflow.net/users/147463 | 375665 | 156,735 |
https://mathoverflow.net/questions/375666 | -3 | Let $X$ be a smooth manifold of dimension $n > 1$. Let us denote by $\underline{\mathbb{S}}^{1}$ the sheaf of the smooth functions over circle, $C^{\infty}$ the sheaf of the smooth functions over $\mathbb{R}$ and by $\underline{\mathbb{Z}}$ the sheaf of the locally constant functions over $\mathbb{Z}$. So, we have an e... | https://mathoverflow.net/users/165195 | Exact sequence of sheaves that generates an exact sequence of Abelian groups | Not sure the question belongs to MO but here you go : if $X = \mathbb C^\*$, consider $f = \frac{z}{|z|} \in C^{\infty}(X, S^1)$. If $f$ is in the image of $exp$, then we found a determination of the logarithm on $S^1$ which is impossible by classical complex analysis.
| 1 | https://mathoverflow.net/users/104742 | 375667 | 156,736 |
https://mathoverflow.net/questions/375649 | 0 | The Erdos-Renyi random graph model G(N,p) describes a way to generate a network with N nodes, the probability that there is a link between any two nodes is p. I am wondering about the probability of generating a ring graph. Based on my understanding, the probability is N!*(p^N)*(1-p)^(N(N-3)). Is it correct?
| https://mathoverflow.net/users/138280 | The probability of generating a ring graph by following the Erdos-Renyi model G(N,p) | Generating precisely a cycle?
There are $(N-1)!$ ways to order the vertices to be in a cycle; a permutation is considered equivalent to another if they generate the same cycle, and each permutation is equivalent to $N$ permutations total including itself. [e.g., on 5 vertices the permutation 12345 is equivalent to 34... | 1 | https://mathoverflow.net/users/122188 | 375679 | 156,739 |
https://mathoverflow.net/questions/375304 | 6 | We have some subsets $A\_1,\dots,A\_k$ of $A=\{1,2,\dots,n\}$. For each permutation $\sigma$ of $A$, define $f(\sigma) = \sum\_{i=1}^k g(\sigma,A\_i)$, where if the earliest element of $A\_i$
in $\sigma$ appears in position $j$, then $g(\sigma,A\_i)= 1/j$. Let $\sigma\_1$ be the permutation maximizing $f(\sigma)$, brea... | https://mathoverflow.net/users/83212 | Permutation function based on subsets | I have a counterexample, showing that $r$ can appear later in $\sigma\_2$ than in $\sigma\_1$.
I'm going to write this counterexample with weights on the sets, but because the weights are rational it can easily be converted to an unweighted counterexample by duplicating sets.
Counterexample: 3x{AC}, 2x{AB}, 2x{BD},... | 3 | https://mathoverflow.net/users/60836 | 375694 | 156,743 |
https://mathoverflow.net/questions/375709 | 6 | A ring $R$ is called semi-perfect if every finitely generated $R$-module has a projective cover, and it can be proved that this is equivalent to say that the category consisting of the finitely generated projective $R$-modules is Krull-Schmidt. I was wondering, and what about the rings $R$ such that every finitely pres... | https://mathoverflow.net/users/nan | Finitely presented modules admitting projective covers | Such rings were called "$F$-semiperfect", and more recently (thanks to rschweib for the information) "semiregular".
One characterization is that these are the rings $R$ such that $\bar{R}=R/J(R)$, the quotient by the Jacobson radical, is von Neumann regular and idempotents lift from $\bar{R}$ to $R$. This is analogou... | 4 | https://mathoverflow.net/users/22989 | 375712 | 156,747 |
https://mathoverflow.net/questions/375715 | 6 | About six years ago there was a proof announced and later outlined in a notice from AMS. However right now I can only seem to find forbidden minor characterizations for matroids linearly representable over $\mathbb{F}\_2,\mathbb{F}\_3,\mathbb{F}\_4$ and some for $\mathbb{F}\_5$. Now understanding the outline given by G... | https://mathoverflow.net/users/38626 | Does the purported proof of Rota's conjecture provide an algorithm for calculating the forbidden minors of matroids over arbitrary finite fields? | As far as I understand, the purported proof does not give an algorithm that given a finite field $\mathbb{F}$, computes the excluded minors for $\mathbb{F}$-representability. This is because it relies on [well-quasi-ordering](https://en.wikipedia.org/wiki/Well-quasi-ordering) arguments, and therefore does not yield exp... | 6 | https://mathoverflow.net/users/2233 | 375717 | 156,749 |
https://mathoverflow.net/questions/366689 | 9 | It is often noted in the literature that there are certain complex periods that allow one to normalize the modular symbol associated to a modular form in such a way that its coefficients are algebraic. This process of normalization by complex periods is regularly attributed to Shimura, though I can't seem to find a con... | https://mathoverflow.net/users/124710 | Proof of a 'well-known' result of Shimura on periods of modular forms | You can find a proof of this theorem (the first in the OP) in Section 5.3 of the following paper by Pasol and Popa: <https://arxiv.org/abs/1202.5802>
The idea is to use the action of Hecke operators. More precisely, the map $f \mapsto \xi\_f^{\pm}$ is Hecke-equivariant, the Hecke operators preserve the rational struc... | 2 | https://mathoverflow.net/users/6506 | 375732 | 156,754 |
https://mathoverflow.net/questions/375729 | 6 | If $R$ is a commutative ring, then the Cayley-Hamilton theorem states that any endomorphism $\phi: R^{n} \rightarrow R^{n}$ of a rank $n$ free module satisfies its own characteristic polynomial, in particular satisfies a monic degree $n$ polynomial.
Now suppose that instead $R = R\_{0} \oplus R\_{1}$ is a commutative... | https://mathoverflow.net/users/16981 | Cayley-Hamilton over super rings | The case when $R$ is purely even but the module has a $Z/2$-grading was studied before, see for example
["On the Cayley-Hamilton equation in the supercase" by Issai Kantor and Ivan Trishin, Comm. in Algebra 27:1 (1999), 233-259](https://www.tandfonline.com/doi/abs/10.1080/00927879908826430)
["Berezinians, Exterior Po... | 6 | https://mathoverflow.net/users/1306 | 375734 | 156,756 |
https://mathoverflow.net/questions/375686 | 3 | The definition of $L$ only permits bounded quantifiers. If we allow a certain number of unbounded quantifiers, does this result in a strict superset of $L$? For example:
$$
\operatorname{Def}^{\Sigma\_3}(X) =
\{ \{ y \mid \text{ $y \in X$ and $\exists x\_1 \forall x\_2 \exists x\_3. (\operatorname{TC}(\{X, x\_1, x\_2... | https://mathoverflow.net/users/65915 | If we have a class like $L$ but allowing a set number of unbounded quantifiers, is it strict superset of $L$? | Every $L^{\Sigma\_n}$, for $n\geq 2$ will be the same as HOD, the class of hereditarily ordinal-definable sets.
This is a consequence of the Myhill-Scott theorem, which asserts that if you form the constructible universe using second-order logic (which means you allow quantifiers over subsets of $X$ only in your set-... | 7 | https://mathoverflow.net/users/1946 | 375742 | 156,758 |
https://mathoverflow.net/questions/375730 | 4 | It seems that whether a simply connected 4 manifold needs 1-handles and 3-handles is still an open question, see [Existence of Morse functions on simply connected manifolds](https://mathoverflow.net/questions/143790/existence-of-morse-functions-on-simply-connected-manifolds).
I am wondering if it is true that every F... | https://mathoverflow.net/users/40517 | Existence of perfect Morse functions on Fermat surfaces $x^n+y^n+z^n+w^n=0$ | Yes, this is true for any nonsingular hypersurface in ${\mathbb C}P^3$. See Harer, On handlebody structures for hypersurfaces in C3 and CP3. Math. Ann. 238 (1978), no. 1, 51–58.
| 5 | https://mathoverflow.net/users/3460 | 375756 | 156,761 |
https://mathoverflow.net/questions/375759 | 19 | In my paper [On the optimal error bound for the first step in the method of cyclic alternating projections](https://arxiv.org/abs/1908.00531), I defined functions $f\_n:[0,1]\to\mathbb{R}$,
$n\geqslant 2$, by
$$
f\_n(c)=\sup\{\|P\_n\dotsm P\_2 P\_1-P\_0\|\,|\,c\_F(H\_1,\dotsc,H\_n)\leqslant c\},
$$
where
(1) the supr... | https://mathoverflow.net/users/48157 | Can we take a supremum over all Hilbert spaces? | It is true that we cannot use an arbitrary property$^1$ $P$ to define a set, in the sense that the collection of **all** things with property $P$ need not be a set. However, the [axiom (scheme) of separation](https://en.wikipedia.org/wiki/Axiom_schema_of_specification) says that we **can** use an arbitrary property to ... | 32 | https://mathoverflow.net/users/8133 | 375762 | 156,765 |
https://mathoverflow.net/questions/375768 | 10 | I apologise in advance if this has been asked here before. I did a search and did not find anything obvious. Erdős' conjecture states that if $A\subseteq {\bf N}$ is such that $\sum\_{n\in A} n^{-1}$ diverges, then $A$ contains arbitrarily long arithmetic sequences.
I was wondering if anything is known about the conv... | https://mathoverflow.net/users/168142 | Converse to Erdős' conjecture on arithmetic progressions | Unfortunately such a simple converse cannot be possible because
one can "plant" long arithmetic progressions in $A$ while keeping it
sparse overall. For example, let $A$ consist of all integers of the form
$10^{n!}+m$ with $1 \leq m \leq n$ (which even makes $\sum\_{n\in A} 1 / \log n$
converge).
[I see that **GH fro... | 17 | https://mathoverflow.net/users/14830 | 375770 | 156,769 |
https://mathoverflow.net/questions/375749 | 0 | Suppose that $\{\mu\_k\}$ is a sequence of probability measures defined on $(I,\mathcal{B})$, where $I$ is a closed real interval and $\mathcal{B}$ is its Borel $\sigma$-algebra, such that:
* each $\mu\_k$ is made of countably many atoms $\{p^n\_k\}$;
* for every $k$, the set $\bigcup\_{n\in\mathbb{N}}p^n\_k$ is not ... | https://mathoverflow.net/users/167834 | Convergence of atomic measures with countably many atoms | (I cannot make any comment so I have no choice but post my comment as answer)
I'm not sure which kind of $\mu\_k$ you want to control. You are surely given tightness for free, so mainly if you can describe the limit law by some total set of linear functionals then your work is done.
I think you know there are a l... | 1 | https://mathoverflow.net/users/168269 | 375773 | 156,770 |
https://mathoverflow.net/questions/372484 | 10 | In general, two conics in the plane intersect at most 4 points. Suppose three of those points are given as $A,B,C$. Then let $c\_1$ be the conic passing through those three points and $D\_1,E\_1$. Let $c\_2$ be the conic passing through those three points and $D\_2,E\_2$. How can the fourth intersection point of these ... | https://mathoverflow.net/users/161614 | Geometric construction of the fourth intersection points of two conics | Based on [*Projective conic sections - constructions*](http://mathafou.free.fr/themes_en/conique7.html),
the crux of the construction is this:
* let two conics intersect in $A,B,C,D$.
* let any line through $A$ intersect the conics again in $M,M'$
* let any line through $B$ intersect the conics again in $N,N'$
* then... | 5 | https://mathoverflow.net/users/5687 | 375775 | 156,771 |
https://mathoverflow.net/questions/375754 | 0 | Suppose we have an operator $A \in B(X,Y)$ where $X$ is a subset of $Y$ (X and Y are Hilbert spaces). Does the fact that $\beta I - A$ is bijective imply that $\beta \in \rho(A)$?
A is closed as an operator from $X$ to $Y$ since it's bounded. Also, for a closed operator $\beta I - A$ bijective implies $\beta \in \rho... | https://mathoverflow.net/users/167962 | Checking if an element belongs to a resolvent set | If $X$ is a Hilbert space with scalar product/norm of $Y$ (that is: complete, and thus closed in $Y$) the argument is fine: In this case $A$ (and equivalently $\beta I-A$) is closed, that is, its graph is closed in $Y\times Y$.
However, I suppose that $X$ is endowed with a different scalar product/norm than $Y$. In t... | 1 | https://mathoverflow.net/users/165275 | 375780 | 156,773 |
https://mathoverflow.net/questions/375778 | 21 | I've been obsessed with this one problem for many months now, and today is the sad day that I admit to myself I won't be able to solve it and I need your help. The problem is simple. We let
$$\mathbb{E}\_{n\in\mathbb{N}}[f(n)]:=\lim\_{N\to\infty}\frac{1}{N}\sum\_{n=1}^{N}f(n)$$
denote the expected value of a functi... | https://mathoverflow.net/users/159298 | Prove or disprove that $\sum_{n=1}^{\infty}\frac{\lambda(n)\mathbb{E}_{n\in\mathbb{N}}[a_n]}{n}=0$ for any choice of $(a_n)$ | This identity is true, though somewhat tricky to prove and the infinite series here might only converge conditionally rather than absolutely.
The key lemma is
>
> **Lemma 1** (Fourier representation of averages along homogeneous arithmetic progressions). Let $a\_n$ be a bounded sequentially summable sequence. The... | 26 | https://mathoverflow.net/users/766 | 375788 | 156,776 |
https://mathoverflow.net/questions/375546 | 9 | Let $k$ be commutative ring and $(C, \Delta)$ be a coalgebra over $k$. Let $D$ be a $k$-submodule of $C$.
Notes I'm reading give the following definition:
>
> $D$ is called subcoalgebra of $C$ if the comultiplication $\Delta: C
> \to C \otimes C$ restricts to a mapping $$\Delta\vert \_D: D \to D
> \otimes D$$
>... | https://mathoverflow.net/users/nan | Definition of subcoalgebra over a commutative ring | I'm surprised nobody has answered yet so let me state (what seems to me to be) the obvious: (2) is the only reasonable definition of sub-coalgebra which makes sense in general. Said differently you want $\Delta:D\rightarrow C\otimes C$ to factor through $D\otimes D \rightarrow C\otimes C$. The definition in your notes ... | 4 | https://mathoverflow.net/users/13552 | 375797 | 156,779 |
https://mathoverflow.net/questions/375805 | 1 | Let $K$ be a number field, and let $C/K$ be a curve of genus $g \geq 2$ over $K$. Suppose that the set of $K$-rational points $C(K)$ on $C$ is non-empty. This enables one to define an *Abel-Jacobi* map $\sigma$ defined over $K$, with a given point $P \in C(K)$, which gives an embedding of $C$ into the Jacobian $J\_C$, ... | https://mathoverflow.net/users/10898 | Algebraic curves and torsion points of its Jacobian | Trivially (though non-trivially according to your definition), we have that $C(K) = \sigma^{-1}(J\_C(K)^{\text{tors}})$ whenever $J\_C(K)$ is finite. There are lots of examples of this kind. But it also should not be very hard to come up with examples of, say, curves of genus 2 over $\mathbb Q$ whose Jacobian has posit... | 7 | https://mathoverflow.net/users/21146 | 375811 | 156,781 |
https://mathoverflow.net/questions/375823 | 3 | This question was motivated by a discussion [here](https://mathoverflow.net/questions/375792/injective-choice-function-for-infinite-complete-linear-hypergraphs) and is related to a previous question [here](https://mathoverflow.net/questions/287380/perfect-matchings-in-infinite-graphs?noredirect=1&lq=1).
Let $\kappa$ ... | https://mathoverflow.net/users/17798 | Perfect matchings in infinite regular bipartite graphs | I believe this is correct (assuming $\lambda\gt0$).
If $\lambda$ is infinite then each connected component of $G$ has $\lambda$ vertices. Since the components can be handled independently, the problem reduces to the $\kappa=\lambda$ case, which can be done by a straightforward transfinite recursion.
If $\lambda$ is... | 5 | https://mathoverflow.net/users/43266 | 375827 | 156,786 |
https://mathoverflow.net/questions/375828 | 5 | Disclaimer: a stronger version of this question was first asked on MSE: <https://math.stackexchange.com/questions/3896547/does-p-nk1-frac1kk-p-n-whenever-0kn/3896842#3896842> and on a French math forum where one person found the counterexample $k=3$ for $n=4$. Still, the person named Ahmad proposed a sketch of proof fo... | https://mathoverflow.net/users/13625 | Does $0<k<n$ imply $p_{n+k}<\left(1+\frac{1}{k}\right)^k p_{n}$ for large enough $n$? | Yes, your inequality, i.e.,
$$p\_{n+k} \lt \left(1 + \frac{1}{k}\right)^{k}p\_n\, , \; \; 0 \lt k \lt n \tag{1}\label{eq1A}$$
does hold for large enough $n$. First, an [approximation for the $n$'th prime number](https://en.wikipedia.org/wiki/Prime_number_theorem), for $n \ge 6$, is
$$n(\log(n) + \log\log(n) - 1) ... | 7 | https://mathoverflow.net/users/129887 | 375836 | 156,790 |
https://mathoverflow.net/questions/375830 | 0 | Does there exist a set $F$ of monotone continuous functions $f \colon [0,1] \to [0,1]$ with the following properties?
1. For each $f \in F$ there exists $x \in [0,1]$ such that $f(x)=1$.
2. There exist $0<m<M<\infty$ such that for all $x \in [0,1)$ and $f \in F$, if $f(x)<1$ then $f$ is $C^2$ at $x$ with
$$ m \ < \ f... | https://mathoverflow.net/users/15570 | Can the identity function be approximated by compositions of a "uniformly monotone-and-convex" set of functions? | No such class $F$ exists, and in fact we do not even need condition 3.
---
First of all, note that every function $f$ in $F$ is non-decreasing and $f(1) = 1$.
Consider the composition $g = f\_n \circ \ldots \circ f\_1$. For every $x \in (0, 1)$, unless $g(x) = 1$, we have
$$ \log g'(x) = \sum\_{i = 1}^n \log f\... | 1 | https://mathoverflow.net/users/108637 | 375846 | 156,794 |
https://mathoverflow.net/questions/375792 | 4 | A [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) $H=(V,E)$ is said to be *complete and linear* if
1. whenever $e\_1\neq e\_2\in E$ then $|e\_1\cap e\_2|=1$, and
2. for $v,w\in V$ there is $e\in E$ such that $\{v,w\}\subseteq e$.
Assuming that $V$ is infinite, is there an injective function $f:E\to V$ such t... | https://mathoverflow.net/users/8628 | Injective choice function for infinite complete linear hypergraphs | After the discussion above, here is what I think is the cleanest proof and it has the property that $f$ is bijection (unless there is an edge of order 1).
If there is an edge of order 1, then we must have $E=\{\{v\}, V\}$ for some $v\in V$, in which case the desired injection is trivial. If there is an edge of order ... | 2 | https://mathoverflow.net/users/17798 | 375847 | 156,795 |
https://mathoverflow.net/questions/375602 | 7 | Szemerédi's theorem states that a strictly increasing sequence of positive integers $a\_0, a\_1, \ldots$ whose range has positive density contains arbitrarily long arithmetic progressions (as subsequences). I was wondering if the statement still holds when the sequence is not required to be strictly increasing (for any... | https://mathoverflow.net/users/168142 | Analogue to Szemerédi's theorem for non-monotone sequences | It appears the statement is false. The paper "On permutations containing no long arithmetic progressions," by Davis, Entringer, Graham, and Simmons [*Acta Arithmetica* **34** (1977)] exhibits a permutation of the positive integers that has no arithmetic progressions of length $5$. The range of this sequence has density... | 2 | https://mathoverflow.net/users/168142 | 375857 | 156,799 |
https://mathoverflow.net/questions/375786 | 0 | Let $Y\_t \in \mathbb{R}$ be a response variable and $X\_t$ a $d$-dimensional explanatory variable. Assume we observe the process that $(X\_1, Y\_1), \cdots, (X\_n, Y\_n)$.
\begin{equation}
Y\_{t} = \mu(X\_{t})+\sigma(X\_{t})\varepsilon\_{t}, \quad t\in\mathbb{Z}
\end{equation}
Then the NW estimator for $\mu(\cdot)$ a... | https://mathoverflow.net/users/153595 | The nonparametric estimation in generalized regression model | The asymptotic expression for $(\hat{m} - m)(x)$ is presented in the Appendix of [Fan and Qao](http://eprints.lse.ac.uk/6635/1/Efficient_estimation_of_conditional_variance_functions_in_stochastic_regression%28LSERO%29.pdf). So this question is meaningless.
| 0 | https://mathoverflow.net/users/153595 | 375865 | 156,802 |
https://mathoverflow.net/questions/375851 | 10 | Let $\mathcal{M}\_r$ be the set of $n \times m$ matrices over $\mathbb{R}$ or $\mathbb{C}$ of rank $r$. What is the Euler characteristic of $\mathcal{M}\_r$? Can someone point me towards a reference for this calculation?
| https://mathoverflow.net/users/152336 | Looking for a reference on the Euler characteristic of the manifold of fixed rank matrices | $\mathcal M\_r$ can be described as a fiber bundle over the product $Gr(r,n) \times Gr(r,m)$ of the Grassmanian of $r$-dimensional subspaces in an $n$-dimensional vector space with the Grassmanian of $r$-dimensional subspaces in an $m$-dimensional vector space, where the fibers are all isomorphic to $GL\_r$. (These are... | 15 | https://mathoverflow.net/users/18060 | 375871 | 156,804 |
https://mathoverflow.net/questions/375395 | 2 | In the introduction to the book Vector bundles and K-theory
[http://pi.math.cornell.edu/~hatcher/VBKT/VBpage.html](http://pi.math.cornell.edu/%7Ehatcher/VBKT/VBpage.html)
two approaches to classification of (topological) vector bundles are discussed - the topological $K$-theory and Characteristic classes. Having al... | https://mathoverflow.net/users/13441 | Calculating topological $K(X)$ for complex projective manifolds | There is an Atiyah-Hirzebruch spectral sequence $H^\*(X;\mathbb{Z}[u,u^{-1}])\Longrightarrow K^\*(X)$, in which the differentials are always torsion-valued. Thus, if $H^\*(X;\mathbb{Z})$ is torsion-free, then the spectral sequence collapses, and $K^\*(X)$ has a natural filtration whose associated graded ring is isomorp... | 5 | https://mathoverflow.net/users/10366 | 375873 | 156,806 |
https://mathoverflow.net/questions/375875 | 2 | It's well known that the following set of real square matrices is dense: those matrices $M$ for which there exists an invertible matrix $P$ such that $P M P^{-1}$ is diagonal. My question is can this statement be strengthened so that $P = P^{-1}$?
The motivation for this question comes from the spectral theorem, whos... | https://mathoverflow.net/users/75761 | Is the following set of real square matrices dense: Those that can be diagonalised by a square root of the identity | **No**
Consider $2\times2$ matrices $M$ whose eigenvalues are not real (they form an open set). The eigenvalues are complex conjugate, as well as the eigenvectors. If $M=P^{-1}DP$, then the columns of $P$ are eigenvectors, thus
$$P=\begin{pmatrix} aw & b\bar w \\ a z & b\bar z \end{pmatrix}.$$
Writing $P^2=I\_2$ give... | 6 | https://mathoverflow.net/users/8799 | 375877 | 156,807 |
https://mathoverflow.net/questions/375874 | 2 | I am looking for a minimal number of properties describing a triangle so that these properties are invariant to the choice of a Cartesian coordinate system as well as to the order in which the triangle points are enumerated. I have a few thousands of data points each of which is a triplet of multidimensional vectors. I... | https://mathoverflow.net/users/124262 | What is the minimum number of triangle centers sufficient to unambiguously describe a triangle? | If the triangle is equilateral, then all well-defined triangle centres coincide, and it's impossible to determine the size of the equilateral triangle.
Otherwise, three triangle centres suffice: the incentre $I$, circumcentre $O$, and Feuerbach point $F$.
In particular, given these three points we can determine:
... | 6 | https://mathoverflow.net/users/39521 | 375880 | 156,808 |
https://mathoverflow.net/questions/372948 | 4 | I am interested in the graphs with treewidth 5 because of its relationship with realization dimension of a graph ([See here](https://link.springer.com/article/10.1007/s00454-006-1284-5)).
In [this PhD thesis](https://smartech.gatech.edu/handle/1853/30061), 75 lists of minimal forbidden minors of a graph with treewidt... | https://mathoverflow.net/users/165923 | Forbidden minors of a graph with treewidth at most 4 | I have a copy of Sander's PhD thesis. Counting $K\_6$, there are actually $76$ excluded minors for treewidth at most $4$ (found by computer) in the thesis, but it is unknown if this list is complete (it is complete up to $8$ vertices). The $76$ graphs are too sporadic to describe here, but send me an email if you want ... | 2 | https://mathoverflow.net/users/2233 | 375882 | 156,809 |
https://mathoverflow.net/questions/375879 | 4 | A similar question was asked before in <https://math.stackexchange.com/questions/2727090/even-perfect-number-that-is-also-a-sum-of-two-cubes>, but no conclusions were drawn.
On the Wikipedia article of [perfect numbers](https://en.wikipedia.org/wiki/Perfect_number#Minor_results) there are two related results concerni... | https://mathoverflow.net/users/168365 | Can an even perfect number be a sum of two cubes? | Here is a proof that 28 is the only even perfect number that is the sum of two positive cubes. The proof in Gallardo's article must be adapted in the case $x,a$ are even.
Write $N=2^{p-1}(2^p-1) = x^3+y^3 = (x+y)(x^2-xy+y^2)$. The gcd $d$ of $x$ and $y$ must be a power of 2, because $d^3$ divides $N$. Writing $x=2^h ... | 6 | https://mathoverflow.net/users/6506 | 375883 | 156,810 |
https://mathoverflow.net/questions/375889 | 2 | On the website of the Berkeley mathematics department there is mention ([see this](https://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=133775)) of a colloquium held on november 5, 2020 (by Zoom) whose speaker was Shinichi Mochizuki, with a talk titled "Classical Roots of Inter-universal Teichmüller The... | https://mathoverflow.net/users/50912 | Berkeley mathematics department colloquium by S.Mochizuki | 1. Yes it really occurred.
2. He was introduced in the normal way, something like "Today we have Shinichi Mochizuki..." that mentioned his name, maybe his talk title and the institution he's from (can't remember exactly.)
3. He annotated those slides with a magnificent glitter-rainbow-colored pen, adding clarificatory ... | 32 | https://mathoverflow.net/users/163893 | 375890 | 156,812 |
https://mathoverflow.net/questions/375893 | 1 | I have a question about higher-order asymptotics of generalized hypergeometric functions. According to <https://dlmf.nist.gov/15.4>
the following is well known:
$$
\_2F\_1(a,b;a+b;z)\sim -\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}\log(1-z),\ \
z\rightarrow1^{-}.
$$
My collaborator was able to coax Wolfram Mathematica into... | https://mathoverflow.net/users/7442 | Higher-order asymptotics of generalized hypergeometric function | In Abramowitz and Stegun, Formula 15.3.11, the equation reads for $m=0,$
$${}\_2F\_1(a,b,a+b) = -\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)} \sum\_{n=0}^\infty
\frac{(a)\_n (b)\_n}{n!^2}(1-z)^n \Big( $$
$$\log(1-z) -2\psi(n+1) + \psi(a+n) + \psi(b+n) \Big)$$
Your asymptotic approximation is the $n=0$ term.
| 2 | https://mathoverflow.net/users/121836 | 375896 | 156,813 |
https://mathoverflow.net/questions/375910 | 5 | Let $\mathfrak A$ be a subset of $\mathrm{Pow}(\mathbb N)$, the powerset of $\mathbb N$. Assume that $\mathfrak A$ is a complete Boolean algebra in the induced order, i.e., the inclusion order. Does it follow that $\mathfrak A$ is atomic?
A complete Boolean algebra $\mathfrak A$ is said to be atomic in case every non... | https://mathoverflow.net/users/66833 | Complete Boolean algebras of subsets of $\mathbb N$ | The answer is negative. Let $A$ be the completion of the denumerable atomless BA $B$. Then $A$ is complete and atomless. $A$ can be isomorphically embedded in $\mathrm{Pow}(\omega)$. In fact, $B$ can be isomorphically embedded in $\mathrm{Pow}(\omega)$, and by Sikorski's extension theorem, this embedding can be extende... | 14 | https://mathoverflow.net/users/168384 | 375914 | 156,818 |
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